Absolutely. Some people are born firefighters. Nothing wrong with that.
I once worked with a senior engineer who loved running incidents. He felt it was real engineering. He loved debugging thorny problems on a strict timeline, getting every engineer in a room and ordering them about, while also communicating widely to the company. Then, there's the rush of the all-clear and the kudos from stakeholders.
Specific to his situation, I think he enjoyed the inflated ownership that the sudden urgency demanded. The system we owned was largely taken for granted by the org; a dead-end for a career. Calling incidents was a good way to get visibility at low-cost, i.e., no one would follow-up on our postmortem action items.
It eventually became a problem, though, when the system we owned was essentially put into maintenance mode, aka zero development velocity. Then I estimate (balancing for other variables) the rate the senior engineer called an incident for not-incidents went up by 3x...
I agree that enjoying firefighting is not inherently harmful. However, the situation you describe afterward irks me in some way I can't quite put my finger on. A lot of words (toxic, dishonest, marketing, counterproductive, bus factor) come to mind, but none of them quite fit.
Some people rise to the occasion during crises and find it rewarding. There's a lot of pop science around COMT (the "warrior gene" associated with stress resilience), which I take with a grain of salt. There does seem to be something there, though, and it overlaps with my personal experience that many great security operations people tend to have ADHD traits.
I've volunteered to fight a share of fires from people who check things in untested, change infrastructure randomly, etc.
What I've learned is that fixing things for these people (and even having entire teams fixing things for weeks) just leads to a continued lax attitude to testing, and leaving the fallout for others to deal with. To them, it all worked out in the end, and they get kudos for rapidly getting a solution in place.
I'm done fixing their work. I'd rather work on my own tasks than fix all the problems with theirs. I'm strongly considering moving on, as this has become an entrenched pattern.